Dormand–Prince method

In numerical analysis, the Dormand–Prince (RKDP) method or DOPRI method, is an embedded method for solving ordinary differential equations (ODE).^[1] The method is a member of the Runge–Kutta family of ODE solvers. More specifically, it uses six function evaluations to calculate fourth- and fifth-order accurate solutions. The difference between these solutions is then taken to be the error of the (fourth-order) solution. This error estimate is very convenient for adaptive stepsize integration algorithms. Other similar integration methods are Fehlberg (RKF) and Cash–Karp (RKCK).

The Dormand–Prince method has seven stages, but it uses only six function evaluations per step because it has the "First Same As Last" (FSAL) property: the last stage is evaluated at the same point as the first stage of the next step. Dormand and Prince chose the coefficients of their method to minimize the error of the fifth-order solution. This is the main difference with the Fehlberg method, which was constructed so that the fourth-order solution has a small error. For this reason, the Dormand–Prince method is more suitable when the higher-order solution is used to continue the integration, a practice known as local extrapolation.^[2]^[3]

Butcher tableau

The Butcher tableau is:

0
1/5	1/5
3/10	3/40	9/40
4/5	44/45	−56/15	32/9
8/9	19372/6561	−25360/2187	64448/6561	−212/729
1	9017/3168	−355/33	46732/5247	49/176	−5103/18656
1	35/384	0	500/1113	125/192	−2187/6784	11/84
	35/384	0	500/1113	125/192	−2187/6784	11/84	0
	5179/57600	0	7571/16695	393/640	−92097/339200	187/2100	1/40

The first row of b coefficients gives the fifth-order accurate solution, and the second row gives the fourth-order accurate solution.

Applications

Dormand–Prince is the default method in the ode45 solver for MATLAB^[4] and GNU Octave^[5] and is the default choice for the Simulink's model explorer solver. It is an option in Python's SciPy ODE integration library^[6] and in Julia's ODE solvers library.^[7] Implementations for the languages Fortran,^[8] Java,^[9] and C++^[10] are also available.

Notes

^ Dormand, J.R.; Prince, P.J. (1980). "A family of embedded Runge-Kutta formulae". Journal of Computational and Applied Mathematics. 6 (1): 19–26. doi:10.1016/0771-050X(80)90013-3.
^ Shampine, Lawrence F. (1986). "Some Practical Runge-Kutta Formulas". Mathematics of Computation. 46 (173): 135–150. doi:10.2307/2008219. JSTOR 2008219.
^ Hairer, Ernst; Wanner, Gerhard; Nørsett, Syvert P. (1993). Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics. Vol. 8. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-540-78862-1. ISBN 978-3-540-56670-0.
^ "Solve nonstiff differential equations — medium order method - MATLAB ode45". www.mathworks.com. Retrieved 2023-08-24.
^ "Matlab-compatible solvers (GNU Octave (version 8.3.0))". octave.org. Retrieved 2023-08-24.
^ "scipy.integrate.RK45 — SciPy v1.11.2 Manual". docs.scipy.org. Retrieved 2023-08-24.
^ "ODE Solvers · DifferentialEquations.jl". docs.sciml.ai. Retrieved 2023-08-24.
^ Hairer, Ernst. "Fortran Codes". www.unige.ch. Retrieved 2023-08-24.
^ "DormandPrince54Integrator (Apache Commons Math 4.0-beta1)". commons.apache.org. Retrieved 2023-08-24.
^ "Class template runge_kutta_dopri5 - 1.53.0". www.boost.org. Retrieved 2023-08-24.

References

Books

Dormand, John R. (1996), Numerical Methods for Differential Equations: A Computational Approach, Boca Raton: CRC Press, pp. 82–84, ISBN 0-8493-9433-3

C. Engstler, C. Lubich (1997), "MUR8: a multirate extension of the eighth-order Dormand-Prince method", Applied Numerical Mathematics, vol. 25, no. 2–3, pp. 185–192, doi:10.1016/S0168-9274(97)00058-5
M. Calvo, J.I. Montijano, L. Randez (1990), "A fifth-order interpolant for the Dormand and Prince Runge-Kutta method", Journal of Computational and Applied Mathematics, vol. 29, no. 1, pp. 91–100, doi:10.1016/0377-0427(90)90198-9{{citation}}: CS1 maint: multiple names: authors list (link)
Jeffrey M. Aristoff, Joshua T. Horwood, Aubrey B. Poore (2014), "Orbit and uncertainty propagation: a comparison of Gauss–Legendre-, Dormand–Prince-, and Chebyshev–Picard-based approaches", Celestial Mechanics and Dynamical Astronomy, vol. 118, no. 1, pp. 13–28, Bibcode:2014CeMDA.118...13A, doi:10.1007/s10569-013-9522-7, S2CID 254378950{{citation}}: CS1 maint: multiple names: authors list (link)
Wo Mei Seen, R. U. Gobithaasan, Kenjiro T. Miura (2014), "GPU acceleration of Runge Kutta-Fehlberg and its comparison with Dormand-Prince method", Germination of Mathematical Sciences Education and Research Towards Global Sustainability (Sksm21), AIP Conference Proceedings, 1605 (1), Penang, Malaysia: 16–21, Bibcode:2014AIPC.1605...16S, doi:10.1063/1.4887558{{citation}}: CS1 maint: multiple names: authors list (link)

[1] Dormand, J.R.; Prince, P.J. (1980). "A family of embedded Runge-Kutta formulae". Journal of Computational and Applied Mathematics. 6 (1): 19–26. doi:10.1016/0771-050X(80)90013-3.

[2] Shampine, Lawrence F. (1986). "Some Practical Runge-Kutta Formulas". Mathematics of Computation. 46 (173): 135–150. doi:10.2307/2008219. JSTOR 2008219.

[3] Hairer, Ernst; Wanner, Gerhard; Nørsett, Syvert P. (1993). Solving Ordinary Differential Equations I. Springer Series in Computational Mathematics. Vol. 8. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/978-3-540-78862-1. ISBN 978-3-540-56670-0.

[4] "Solve nonstiff differential equations — medium order method - MATLAB ode45". www.mathworks.com. Retrieved 2023-08-24.

[5] "Matlab-compatible solvers (GNU Octave (version 8.3.0))". octave.org. Retrieved 2023-08-24.

[6] "scipy.integrate.RK45 — SciPy v1.11.2 Manual". docs.scipy.org. Retrieved 2023-08-24.

[7] "ODE Solvers · DifferentialEquations.jl". docs.sciml.ai. Retrieved 2023-08-24.

[8] Hairer, Ernst. "Fortran Codes". www.unige.ch. Retrieved 2023-08-24.

[9] "DormandPrince54Integrator (Apache Commons Math 4.0-beta1)". commons.apache.org. Retrieved 2023-08-24.

[10] "Class template runge_kutta_dopri5 - 1.53.0". www.boost.org. Retrieved 2023-08-24.

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